The asymptotic of products of general Markov/transition kernels is investigated using Doeblin’s coefficient. We propose a general approximating scheme as well as a convergence rate in total variation of such products by a sequence of positive measures. These approximating measures and the control of convergence are explicit from the two parameters in the minorization condition associated with the Doeblin

In this paper, we study the speed of mixing of absorbing Markov processes with state space E, that is, the rate of the convergence in the following limits: limt→∞Ex[f(Xpt)g(Xt)|T>t]=∫Ef(y)β(dy)∫Eg(y)α(dy), limt→∞Ex[f(Xpt)g(Xqt)|T>t]=∫Ef(y)β(dy)∫Eg(y)β(dy),where f,g are bounded and measurable functions defined on E, x∈E, 0

We introduce a simple criterion for studying stationarity and moments properties of some multivariate Markovian autoregressive processes, under a contracting mapping assumption. We apply our results to the Poisson INGARCH model and to one of its multivariate extension recently introduced in the literature. In particular, we obtain optimal stationarity conditions and existence of some exponential moments

Summary: We construct μp-optimal approximate designs for comparing population curves in hierarchical models with group-specific treatment. Equivalence theorems are given to confirm μp-optimality of designs and the optimal allocation of sample size in each group. An illustrative application to find μ1 and μ∞-optimal designs is presented for the random slope models with two-treatment group assignments

A known property of conditional expectation is extended to the framework of Markov kernels. Its meaning in terms of densities is provided. Some examples located in the field of clinical diagnosis are presented to delimit the main result of the paper.

In this note we prove a large deviation result for the intersection of ranges of two independent random walks in dimension two. This complements the study of Phetpradap from 2011, where the intersection in dimension three and above was studied.

Let {Zn} be a supercritical branching process in an independent and identically distributed random environment. As is well known, the behavior of Zn depends primarily on that of the associated random walk Sn constructed by the logarithms of the quenched expectation of population sizes. By this observation, the Berry–Esséen bound for logZn has been established by Grama et al. (2017). To refine that

Recently, Chen, Li and Zhang established conditions characterizing asymptotic identifiability of latent factors in confirmatory factor analysis. We give an elementary proof showing that a similar characterization holds non-asymptotically, and prove a related result for identifiability of factor loadings.

In this paper, we consider the slow-fast stochastic systems with singular coefficients. Using Zvonkin’s transformation and the strong convergence in the averaging principle, we establish quantitative stability estimates both for the original systems and the corresponding averaged equations.

This paper is concerned with the exponential contraction of Markovian switching jump diffusions that involve a partially observed Markov chain. For such partially observed problems, we use a Wonham filter to estimate the Markov chain from the observable evolution of the given process, and convert the original system to a completely observable one. We then establish sufficient conditions of feedback

Without higher moment assumptions, this note establishes the decay of the Kolmogorov distance in a central limit theorem for Lévy processes. This theorem can be viewed as a continuous-time extension of the classical random walk result by Friedman etal. (1966).

This paper studies the unconditional limiting distribution of the maximal deviation of bootstrap kernel density estimators with re-sample sizes that are different from the sample size, n. More specifically, we study the convergence rates of such statistics when the bootstrap sample size may be orders of magnitude smaller than n. An application to big-data scenarios is given.

This paper develops optimal statistical tests for testing certain class of non-linear time series models contiguous to a non-linear autoregressive processes with β-ARCH errors. The statistical tests are based on the Local Asymptotic Normality -LAN-(established via the quadratic mean differentiability) of the log-likelihood ratio of the studied Model when its parameters are estimated. Their local power

We derive the asymptotic rate of decay to the tail dependence coefficient, zero, of the bivariate Variance-Gamma distribution under an equal-skewness condition, using the bivariate skew Generalized Hyperbolic distribution. The bivariate problem is first reduced to a univariate one.

This paper deals with the problem of nonparametric relative error regression function estimation for twice censored data. A new estimate is proposed, which is built by combining the local linear approach and relative error estimation. A uniform almost sure consistency with rate over a compact set is established. A numerical study is carried out to assess the performance of the proposed estimator. Practical

In this correspondence, we present new results concerning the concept of stochastic domination and apply them to obtain new results on uniform integrability and on the strong law of large numbers for sequences of pairwise independent random variables. Our result on the strong law of large numbers extends a result of Chen, Bai, and Sung (2014). The sharpness of the results is illustrated by three examples

This paper proposes a new correlation ratio based on the bilinear extension copula regression to describe asymmetric association for ordinal contingency tables. Theoretical and finite sample properties of the proposed measure and its estimator are studied.

Douma et al. (2018) introduced a recursive kernel estimator of the hazard function in the framework of independent rightly censored data, and studied its weak convergence rate, without giving any result on its strong behavior. The aim of this paper is to establish a compact law of the iterated logarithm for this estimator, which requires an approach in the proof totally different from that used in

This paper is devoted to the conditional density estimation for continuous time processes with values in functional spaces. Under standard assumptions, we prove the uniform convergence of the conditional density estimator, from which we deduce the almost sure convergence of the conditional mode estimator. Then, the convergence of the conditional distribution function and the regression function estimators

While sparse factor regression is often encountered under high dimensions, it still is unclear how to control the false discovery rate (FDR) of latent factors. In this paper, we propose a variable selection procedure to address the issue and prove that the FDR can be asymptotically controlled at a target level. Moreover, our approach is scalable and memory-efficient in practice owing to the divide-and-conquer

Estimation method for Hurst indices in operator scaling Gaussian random field is developed. The model used in this paper has two Hurst parameters along the two orthogonal directions. The two directions are estimated first, then Hurst indices are estimated along the estimated directions. The performance of estimator is investigated theoretically and empirically.

We introduce here a generalization of the Mittag-Leffler Lévy process (with parameter α), obtained by extending its Lévy measure through the Prabhakar function (which is a Mittag-Leffler with the additional parameters β and γ). We prove that this so-called Prabhakar process, in the special case β=1, can be represented as an α-stable process subordinated by an independent generalized gamma subordinator;

Using the method of Girsanov’s transformation, we investigate Talagrand’s quadratic transportation cost inequalities for the law of the solution of backward stochastic differential equations with mean reflection, under the uniform norm and L2-norm. These equations are driven by a Brownian motion.

Let X, Y be càdlàg martingales and let Y# denote the sharp function of Y. The paper contains the proof of the estimate ‖∫0∞|d〈X,Y〉t|‖1≤2‖〈X〉1∕2Y#‖1 for the total variation between X and Y. The constant 2 is shown to be the best possible. The proof rests on the construction of an appropriate special function, enjoying certain size and concavity requirements.

In the paper, we are concerned with degenerate stochastic differential equations with jumps. We first establish two theorems about supports for the solution laws of the degenerate stochastic differential equations, under different (sufficient) conditions. We then apply one of our results to a class of degenerate stochastic evolution equations (that is, stochastic differential equations in infinite

Unmeasured confounding is a threat to causal inference and individualized decision making. Similar to Cui and Tchetgen Tchetgen (2020); Qiu et al. (2020); Han (2020a), we consider the problem of identification of optimal individualized treatment regimes with a valid instrumental variable. Han (2020a) provided an alternative identifying condition of optimal treatment regimes using the conditional Wald

In this paper we will give a probabilistic representation for the heat flow of harmonic map with time-dependent Riemannian metric via a forward–backward stochastic differential equation on Riemannian manifolds. Moreover, we can provide an alternative stochastic method for the proof of existence of a unique local solution for heat flow of harmonic map with time-dependent Riemannian metric.

We consider here the problem of estimating the p×p scale matrix Σ of a multivariate linear regression model when the distribution of the observed matrix belongs to a large class of elliptically symmetric distributions. Any estimator Σˆ of Σ is assessed through the data-based loss tr(S+Σ(Σ−1Σˆ−Ip)2)  where S is the sample covariance matrix and S+ is its Moore–Penrose inverse.

This paper concerns the minimax estimation of covariance and precision matrices for high-dimensional time series with long-memory property. We generalize the minimax results for the convergence rates of the estimation of covariance matrices in Shu and Nan (2019) in several directions with a mild assumption, which was mentioned as an open problem in Supplement to Cai and Zhou (2012) for i.i.d. data

In this paper we consider a concept of statistical causality, based on Granger’s definition of causality and analyze the relationships between given causality and the concept of measurable separability of σ-algebras. The measurable separability of σ-algebras is defined in Florens et al. (1990). We give a generalization of that definition for flows of information represented by filtrations and consider

We propose a test for the null of strict stationarity in a Random Coefficient AutoRegression (RCAR) of order 1. The test can also be used in the case of a standard AR(1) model, and it can be applied under minimal requirements on the existence of moments — in both cases without requiring any modifications or prior knowledge.

We propose a recursive method for the computation of the cumulants of self-exciting point processes of Hawkes type, based on standard combinatorial tools such as Bell polynomials. This closed-form approach is easier to implement on higher-order cumulants in comparison with existing methods based on differential equations, tree enumeration or martingale arguments. The results are corroborated by Monte

In this note we proved that weak limits, of sums of positive, independent and identically distributed random variables which are re-normalized by a non-linear shrinking transform max(0,x−r), are either degenerate or (some) compound Poisson distributions.

This paper considers a nonstandard renewal risk model with constant force of interest, in which each main claim induces a delayed claim. Assume that the main claim sizes and the inter-arrival times obey a dependence structure, and so do the delayed claim sizes and the arrival times. Supposing that the distributions of the main and delayed claim sizes both have subexponential tails, asymptotic estimate

For bivariate random lifetimes, we study the role of stochastic dependence through conducting stochastic comparison on the residual life, inactivity time and some aging behavior of one marginal conditioned by the survival or failure of the other. The main results serve as essential supplement to those related ones of Bassan and Spizzichino (1999) and Longobardi and Pellerey (2019). Applications in

Panel data models can be found in many areas such as economics, social science, medicine, and epidemiology. In the last decade, the inference of regression coefficients in panel data models has seen development. In this paper, we introduce three pivotal quantities to implement an equality hypothesis on the regression coefficients vectors. We show that our newly proposed methods have advantages over

The direct Gaussian copula model with discrete margins is appealing but poses computational challenges due to its intractable likelihood. We show that the distributional transform-based approximate likelihood is essentially exact for some variants of the model, and we propose a quantity that can be used to assess exactness for a given dataset.

In this article, we prove two new versions of a theorem proven by Efron in Efron (1965). Efron’s theorem says that if a function ϕ:R2→R is non-decreasing in each argument then we have that the function s↦E[ϕ(X,Y)|X+Y=s] is non-decreasing. We name restricted Efron’s theorem a version of Efron’s theorem where ϕ:R→R only depends on one variable. PFn is the class of functions such as ∀a1≤⋯≤an,b1≤⋯≤bn,det(f(ai−bj))1≤i

Three common classes of kernel regression estimators are considered: the Nadaraya–Watson (NW) estimator, the Priestley–Chao (PC) estimator, and the Gasser–Müller (GM) estimator. It is shown that (i) the GM estimator has a certain monotonicity preservation property for any kernel K, (ii) the NW estimator has this property if and only the kernel K is log concave, and (iii) the PC estimator does not have

Let u(t,x) be the solution to the stochastic heat equation on R+×Rd driven by a Gaussian noise that is white in time and has a spatially covariance that satisfies Dalang’s condition. In this paper, we prove an almost sure central limit theorem for spatial averages of the form ∫[0,N]dg(u(t,x))dx as N→∞ for fixed t>0, where g is a globally Lipschitz function or belongs to a class of locally Lipschitz

For a binomial random variable ξ with parameters n and b∕n, it is well known that the median equals b when b is an integer. In 1968, Jogdeo and Samuels studied the behaviour of the relative difference between P(ξ=b) and 1∕2−P(ξ

Chao estimator is not guaranteed to be asymptotically conservative with finite sampling efforts. A simple correction solves the issue. We illustrate with simulations and examples that the corrected Chao estimator is asymptotically conservative, and has lower standard error.

Let (ξ1,η1), (ξ2,η2),… be independent and identically distributed R2-valued random vectors. Put S0≔0 and Sk≔ξ1+…+ξk for k∈N. We prove a functional central limit theorem for a discounted exponential functional of the random walk ∑k≥0e−Sk∕t, properly normalized and centered, as t→∞. In combination with a theorem obtained recently in Iksanov et al. (2021) this leads to an ultimate functional central limit

The paper presents an omnibus goodness-of-fit (Gof) test for renewal processes based on Dawid’s prequential framework. This test compares models on the basis of their predictive abilities. It uses usual Edf statistics (KS, AD or CvM) of a prequential empirical process which, under standard regulatory conditions, is proven to converge to the Brownian bridge as in the case of known parameters. This result

In this note, a general approach to the study of non-stationary Markov chains with catastrophes and the corresponding queuing models is considered, as well as to obtain estimates of the limiting regime itself. As an illustration, an example of a queuing model is studied.

In this paper, a new monotonic limit theorem for regulated processes is developed, from that proved by Peng (1999). This theorem is a useful tool to prove the existence theorem of reflected backward stochastic differential equation when the barrier has regulated trajectories (not necessarily right-continuous) by means of penalization method.

It is well known that there are two regimes in a standard one-dimensional Boolean percolation model: either the entire space is covered a.s., or the covered volume fraction is strictly less than one. The aim of this work is to demonstrate that there is a third possibility in a Boolean model with totally asymmetric grains on a half-line: a.s. there is no unbounded component, but the covered volume fraction

Shrinkage estimation of eigenvalues of covariance matrices for Gaussian spiked covariance models are known to be effective especially in high-dimensional problems. Various shrinkage methods have been proposed, and these typically employ nonlinear functions of eigenvalues of sample covariance matrices. Here we investigate Bayesian shrinkage methods for single-spiked covariance models. The choice of

A probability distribution μ on Rd is quasi-infinitely divisible if its characteristic function has the representation μ̂=μ1̂∕μ2̂ with infinitely divisible distributions μ1 and μ2. In Lindner et al. (2018, Thm. 4.1) it was shown that the class of quasi-infinitely divisible distributions on R is dense in the class of distributions on R with respect to weak convergence. In this paper, we show that the

We consider the determinacy problem for a distribution function F(x), x∈R, by mixed data consisting of partial information given on F and knowing its finitely many moments. Necessary and sufficient conditions for solvability of the problem are derived. Essential is the established one-to-one correspondence between the Hamburger moment problem with finitely many moments and a discrete positive Borel

For a sequence of independent and identically distributed random variables, Jajte (2003) established a strong law of large numbers for weighted sums of the random variables. In this paper, we generalize the result of Jajte (2003). We also obtain a new strong law of large numbers for weighted sums of the random variables.

We present a relatively simple and mostly elementary proof of the Lévy–Khintchine formula for subordinators. The main idea is to study the Poisson process time-changed by the subordinator. This is a compound Poisson process which is easy to investigate using elementary probabilistic techniques. It turns out that its rate equals the value of the Laplace exponent of the leading subordinator at 1, and

Continuity of the value of the martingale optimal transport problem on the real line w.r.t. its marginals was recently established in Backhoff-Veraguas and Pammer (2019) and Wiesel (2019). We present a new perspective of this result using the theory of set-valued maps. In particular, using results from Beiglböck et al. (2021), we show that the set of martingale measures with fixed marginals is continuous

A sequence of independent Bernoulli trials, each of which is a success with probability p, is conducted. For k∈Z+, let Xk be the number of trials required to obtain k consecutive successes. Using techniques from elementary probability theory, we present a derivation which ultimately yields an elegant expression for the probability mass function of Xk, and is simpler in comparison to what is found in

In this paper, we mainly study the recursive density estimators of the probability density function for strong mixing random variables. The rate of complete consistency is established under some suitable conditions. As an application, we further investigate the rate of complete consistency for hazard rate function estimator. We also present some numerical studies to verify the validity of our theoretical

In this paper, the limit properties of properly time-scaled and normalized maxima of minimum of vector-valued Gaussian processes are studied. It is shown that the maxima of dependent samples of those processes converge weakly on the space of continuous functions to a stochastic process with explicit finite-dimensional distributions.

Herein, we address the expectations of frame potentials of three types of determinantal point processes (DPPs) on the unit sphere Sd: (i) spherical ensembles on S2; (ii) harmonic ensembles on Sd and (iii) jittered sampling point processes on Sd. The random point configurations generated by such DPPs converge more rapidly towards finite unit norm tight frames (FUNTFs) than the Poisson point processes

The asymptotic variance is a natural indicator of the efficiency for a Markov Chain Monte Carlo algorithm. In this note, we prove that the asymptotic variance of a drift accelerated diffusion converges to zero uniformly if and only if there are no non-trivial first order Sobolev functions in the kernel of the drift generating operator. Its proof is based on spectral analysis in the first order Sobolev

We describe the class of functions f:Rn→Rm which transform a vector Brownian Motion into a martingale and use this description to give martingale characterization of the general measurable solution of the multidimensional Cauchy functional equation.

This paper provides a partial solution to a long-standing open problem posted by Balakrishnan and Zhao (2013) concerning the likelihood ratio ordering of the lifetimes of parallel systems with two heterogeneous exponential components. The proposed method can be applied to tackle other stochastic ordering problems in a parallel system.